Yukawa potentials in systems with partial periodic boundary conditions I : Ewald sums for quasi-two dimensional systems

TL;DR

This paper derives the Ewald summation method for Yukawa potentials in systems with partial (quasi-two-dimensional) periodic boundary conditions, applicable for any Debye length, including the Coulomb limit, improving accuracy in simulations.

Contribution

It provides a new, closed-form derivation of Ewald sums for Yukawa potentials in systems with two-dimensional periodicity, extending previous work to partial boundary conditions.

Findings

Derived Ewald sums for quasi-two-dimensional Yukawa systems.

Clarified the Coulomb limit and electroneutrality relation.

Applicable for any Debye length in simulations.

Abstract

Yukawa potentials are often used as effective potentials for systems as colloids, plasmas, etc. When the Debye screening length is large, the Yukawa potential tends to the non-screened Coulomb potential ; in this small screening limit, or Coulomb limit, the potential is long ranged. As it is well known in computer simulation, a simple truncation of the long ranged potential and the minimum image convention are insufficient to obtain accurate numerical data on systems. The Ewald method for bulk systems, i.e. with periodic boundary conditions in all three directions of the space, has already been derived for Yukawa potential [cf. Y., Rosenfeld, {\it Mol. Phys.}, \bm{88}, 1357, (1996) and G., Salin and J.-M., Caillol, {\it J. Chem. Phys.}, \bm{113}, 10459, (2000)], but for systems with partial periodic boundary conditions, the Ewald sums have only recently been obtained [M., Mazars, {\it J. Chem. Phys.}, {\bf 126}, 056101 (2007)]. In this paper, we provide a closed derivation of the Ewald sums for Yukawa potentials in systems with periodic boundary conditions in only two directions and for any value of the Debye length. A special attention is paid to the Coulomb limit and its relation with the electroneutrality of systems.